STABILITY ANALYSIS OF A RATIO-DEPENDENT
PREDATOR-PREY SYSTEM WITH DIFFUSION
AND STAGE STRUCTURE
XINYU SONG, ZHIHAO GE, AND JINGANG WU
Received 16 November 2004; Revised 25 January 2006; Accepted 16 February 2006
A two-species predator-prey system with diffusion term and stage structure is discussed,
local stability of the system is studied using linearization method, and global stability of
the system is investigated by strong upper and lower solutions. The asymptotic behavior
of solutions and the negative effect of stage structure on the permanence of populations
are given.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Predator-prey models have been studied by many authors (see [6, 21]), but the stage
structure of species has been ignored in the existing literature. In the natural world, however, there are many species whose individual members have a life history that take them
through two stages: immature and mature (see [1–3, 7–9, 18–20]). In particular, we have
in mind mammalian populations and some amphibious animals, which exhibit these two
stages. In these models, the age to maturity is represented by a time delay, leading to a system of retarded functional differential equations. For general models one can see [11].
Specifically, the standard Lotka-Volterra type models, on which nearly all existing
theories are built, assume that the per capita rate predation depends on the prey numbers only. An alternative assumption is that, as the numbers of predators change slowly
(relative to prey change), there is often competition among the predators and the per
capita rate of predation depends on the numbers of both preys and predators, most likely
and simply on their ratio. A ratio-dependent predator-prey model has been investigated
by [10].
Recently, a model of ratio-dependent two species predator-prey with stage structure
was derived in [19]. The model takes the form
dX1 (t)
= αX2 (t) − γX1 (t) − αe−γτ X2 (t − τ),
dt
dX2 (t)
cX2 (t)Y (t)
= αe−γτ X2 (t − τ) − βX2 2 (t) −
,
dt
X2 (t) + mY (t)
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 13948, Pages 1–20
DOI 10.1155/IJMMS/2006/13948
2
A ratio-dependent predator-prey system
f X2 (t)
dY (t)
= Y (t) − d +
,
dt
X2 (t) + mY (t)
x1 (0) > 0,
x2 (t) = ϕ(t) ≥ 0,
y(0) > 0,
−τ ≤ t ≤ 0,
(1.1)
where X1 (t), X2 (t) represent, respectively, the immature and mature prey populations
densities; Y (t) represents the density of predator population; f > 0 is the transformation
coefficient of mature predator population; αe−γτ X2 (t − τ) represents the immatures who
were born at time t − τ and survive at time t (with the immature death rate γ), and
τ represents the transformation of immatures to matures; α > 0 is the birth rate of the
immature prey population; γ > 0 is the death rate of the immature prey population; and
β > 0 represents the mature death and overcrowding rate. The model is derived under the
following assumptions.
(H1) The birth rate of the immature prey population is proportional to the existing
mature population with a proportionality constant α > 0; the death rate of the
immature prey population is proportional to the existing immature population
with a proportionality constant γ > 0; we assume for the mature population that
the death rate is of a logistic nature.
(H2) In the absence of prey spaces, the population of the predator decreased, and d > 0,
f > 0, m > 0.
Note that the first equation of system (1.1) can be rewritten to
X1 (t) =
t
t −τ
αe−γ(t−s) X2 (s)ds,
(1.2)
so we have
X1 (0) =
0
−τ
αeγs X2 (s)ds.
(1.3)
This suggests that if we know the properties of X2 (t), then the properties of X1 (t) can be
obtained from X2 (t) and Y (t). Therefore, in the following we need only to consider the
following model:
cX2 (t)Y (t)
dX2 (t)
= αe−γτ X2 (t − τ) − βX2 2 (t) −
,
dt
X2 (t) + mY (t)
f X2 (t)
dY (t)
= Y (t) − d +
,
dt
X2 (t) + mY (t)
x1 (0) > 0,
y(0) > 0,
x2 (t) = ϕ(t) ≥ 0,
(1.4)
−τ ≤ t ≤ 0.
In [19], the effect of delay on the populations and the global asymptotic attractivity
of the system (1.4) were considered, for detailed results we refer to [19]. However, the
diffusion of the species which is in addition to the species’ natural tendency to diffuse to
areas of smaller population concentration is not considered. For the details of diffusion
in different areas, we can see [4, 12–17, 22]. In this paper, we study the system (1.1) with
Xinyu Song et al. 3
diffusion terms, taking into account the diffusion of the species in different areas. The
role of diffusion in the following system of nonlinear pdes with diffusion terms and stage
structure will be studied:
∂u1
− D1 Δu1 = αu2 (x,t) − γu1 (x,t) − αe−γτ u2 (x,t − τ),
∂t
cu2 (x,t)v(x,t)
∂u2
− D1 Δu2 = αe−γτ u2 (x,t − τ) − βu2 2 (x,t) −
,
∂t
u2 (x,t) + mv(x,t)
f u2 (x,t)
∂v
− D2 Δv = v(x,t) − d +
,
∂t
u2 (x,t) + mv(x,t)
∂u1 ∂u2 ∂v
=
=
= 0,
∂n
∂n
∂n
u1 (x,t) = ϕ1 (x,t),
u2 (x,t) = ϕ2 (x,t),
x ∈ Ω, t > 0,
(1.5)
x ∈ ∂Ω, t > 0,
x ∈ Ω̄, t ∈ [−τ,0],
v(x,0) = ϕ3 (x,0),
where ∂/∂n is differentiation in the direction of the outward unit normal to the boundary
∂Ω, we assume Ω ⊂ RN is open, bounded and ∂Ω is smooth. The diffusion coefficients D1 ,
D2 , and D3 are positive. The homogeneous Neumann boundary condition indicates that
the predator-prey system is self-contained with zero population flux across the boundary.
The initial functions ϕ1 (x,t), ϕ2 (x,t), and ϕ3 (x,t) are Hölder continuous, and satisfy the
compatible condition
∂ϕi
= 0 on ∂Ω, i = 1,2,3.
∂n
(1.6)
Denote u2 (x,t) and v(x,t) as u1 (x,t) and u2 (x,t), respectively, so we get the following
subsystem of the system (1.5):
∂u1
cu1 (x,t)u2 (x,t)
− D1 Δu1 = αe−γτ u1 (x,t − τ) − βu1 2 (x,t) −
,
∂t
u1 (x,t) + mu2 (x,t)
f u1 (x,t)
∂u2
− D2 Δu2 = u2 (x,t) − d +
,
∂t
u1 (x,t) + mu2 (x,t)
∂u1
= 0,
∂n
u1 (x,t) = ϕ1 (x,t),
∂u2
= 0,
∂n
x ∈ Ω, t > 0,
(1.7)
x ∈ ∂Ω, t > 0,
u2 (x,t) = ϕ2 (x,0),
x ∈ Ω̄, t ∈ [−τ,0].
Note that the quantities u2 (x,t) and v(x,t) of the system (1.5) are independent of the
quantity u1 (x,t), so we may only consider the subsystem (1.7) to be easy to get the properties of the system (1.5).
Before proceeding further, let us nondimensionalize the system (1.7) with the following scaling: U1 = βu1 , U2 = mβu2 , T = t, by rewriting U1 , U2 , T to u1 , u2 , t, respectively.
4
A ratio-dependent predator-prey system
We obtain the following nondimensionless system:
bu1 (x,t)u2 (x,t)
∂u1
− D1 Δu1 = au1 (x,t − τ) − u1 2 (x,t) −
,
∂t
u1 (x,t) + u2 (x,t)
f u1 (x,t)
∂u2
− D2 Δu2 = u2 (x,t) − d +
,
∂t
u1 (x,t) + u2 (x,t)
∂u1
= 0,
∂n
u1 (x,t) = ϕ1 (x,t),
∂u2
= 0,
∂n
x ∈ Ω, t > 0,
(1.8)
x ∈ ∂Ω, t > 0,
u2 (x,t) = ϕ2 (x,0),
x ∈ Ω̄, t ∈ [−τ,0],
where a = αe−γτ , b = c/m.
The remaining part of this paper is organized as follows. The existence and uniqueness of the solutions of system (1.8) will be proved in Section 2. In Section 3, we obtain
conditions for local asymptotic stability of the nonnegative equilibria of system (1.8).
In Section 4, we analyze the global asymptotic stability and obtain conditions for global
asymptotic stability of the nonnegative equilibria of system (1.8).
2. Existence and uniqueness of the solutions
In order to solve the problem and prove theorems, we devote to some preliminaries. We
rewrite system (1.8) to
∂u1
− D1 Δu1 = F1 u1 (x,t),u2 (x,t),u1 (x,t − τ) ,
∂t
∂u2
− D2 Δu2 = F2 u1 (x,t),u2 (x,t) ,
∂t
∂u1
= 0,
∂n
u1 (x,t) = ϕ1 (x,t),
∂u2
= 0,
∂n
x ∈ Ω, t > 0,
(2.1)
x ∈ ∂Ω, t > 0,
u2 (x,0) = ϕ2 (x,0),
x ∈ Ω̄, t ∈ [−τ,0],
where F1 (u1 (x,t),u2 (x,t),u1 (x,t − τ)) = au1 (x,t − τ) − u1 2 (x,t) − bu1 (x,t)u2 (x,t)/
(u1 (x,t) + u2 (x,t)), and F2 (u1 (x,t),u2 (x,t)) = u2 (x,t)(−d + f u1 (x,t)/(u1 (x,t) + u2 (x,t))).
Definition 2.1. Suppose ϕ1 (x,t), ϕ2 (x,t), ψ(x,t) be Hölder continuous, call (u1 , u2 ),
(u
1 , u
2 ) to be a pair of strong upper and lower solutions, if u1 , u
1 , u2 , and u
2 ∈ C(Ω̄ ×
[0,+∞)) ∩ C 2,1 (Ω × [0,+∞)) such that u
1 ≤ u1 , u
2 ≤ u2 , and
∂u1
bu1 (x,t)u
2 (x,t)
− D1 Δu
1 ≥ au
1 (x,t − τ) − u
21 (x,t) −
,
∂t
u1 (x,t) + u
2 (x,t)
bu
1 (x,t)u2 (x,t)
∂u
1
− D1 Δu
1 ≤ au
1 (x,t − τ) − u
21 (x,t) −
,
∂t
u
1 (x,t) + u2 (x,t)
f u1 (x,t)u2 (x,t)
∂u2
− D2 Δu
2 ≥ −d u
2 (x,t) +
,
∂t
u1 (x,t) + u2 (x,t)
Xinyu Song et al. 5
f u
1 (x,t)u
2 (x,t)
∂u
2
− D2 Δu
2 ≤ −d u
2 (x,t) +
,
∂t
u
1 (x,t) + u
2 (x,t)
∂u1
∂u
1
≤0≤
,
∂n
∂n
∂u2
∂u
2
≤0≤
,
∂n
∂n
u
1 (x,t) ≤ ϕ1 (x,t) ≤ u1 (x,t),
x ∈ Ω, t > 0,
(x,t) ∈ ∂Ω × [0,+∞),
(x,t) ∈ Ω̄ × [−τ,0],
u
2 (x,0) ≤ ϕ2 (x,0) ≤ u2 (x,0),
x ∈ Ω̄.
(2.2)
Similar to Definition 2.1, the definition of a pair of strong upper and lower solutions
of the elliptic system corresponding to system (2.1) is easy to be given.
Lemma 2.2 [14]. Suppose that ui (x,t) ∈ C(Ω̄ × [0,T]) ∩ C 2,1 (Ω × [0,T]) satisfy
∂ui
− Di Δui ≥
bi j u j (x,t) + ci j u j x,t − τi ,
∂t
j =1
j =1
2
2
∂ui
≥ 0, (x,t) ∈ ∂Ω × [0,T];
∂n
(x,t) ∈ Ω × [0,T],
(2.3)
ui (x,t) ≥ 0, (x,t) ∈ Ω × [−τ,0],
where bi j (x,t),ci j (x,t) ∈ C(Ω̄ × [0,T]), bi j ≥ 0 for (i = j), and ci j ≥ 0 for i, j = 1,2, and
τ2 = 0. Then
ui (x,t) ≥ 0,
(x,t) ∈ Ω̄ × [0,T].
(2.4)
From Lemma 2.2 we easily get the following lemma.
Lemma 2.3. For any given T > 0, if u(x,t) and v(x,t) belong to C(Ω̄ × [0,T]) ∩ C 2,1 (Ω ×
[0,T]) and satisfy the relations
∂u
− DΔu − au(x,t − τ) − βu2 (x,t)
∂t
∂v
− DΔv − av(x,t − τ) − βv 2 (x,t) ,
≥
∂t
∂u ∂v
≥
,
∂n ∂n
x ∈ ∂Ω, t ∈ [0,T];
x ∈ Ω, t ∈ [0,T],
u(x,t) = ϕ(x,t) ≥ v(x,t),
x ∈ Ω̄, t ∈ [−τ,0].
(2.5)
Then u(x,t) ≥ v(x,t).
Proof. Let ω(x,t) = u(x,t) − v(x,t), then
∂ω
− DΔω ≥ au(x,t − τ) − βu2 (x,t) − av(x,t − τ) − βv 2 (x,t)
∂t
= aω(x,t − τ) − βω(x,t) u(x,t) + v(x,t) .
(2.6)
6
A ratio-dependent predator-prey system
Let c11 = a, b11 = −β(u(x,t) + v(x,t)). Since c11 = a = αe−γτ > 0, by Lemma 2.2 we have
ω(x,t) ≥ 0, that is,
u(x,t) ≥ v(x,t).
(2.7)
Theorem 2.4. Let u1 (x,t) and u2 (x,t) be the solutions of system (2.1) in C(Ω̄ × [0,T]) ∩
C 2,1 (Ω × [0,T]), and if f > d, then
def
0 ≤ u1 (x,t) ≤ max ϕ1 ∞ ,a = M1 ,
M1 ( f − d)
0 ≤ u2 (x,t) ≤ max ϕ2 ∞ ,
def
(2.8)
= M2 .
d
Proof. Let 0 ≤ σ ≤ T. In order to investigate system (2.1), we firstly consider the following
system:
∂ψ1
− D1 Δψ1 = aψ1 (x,t − τ) + ψ1 (x,t) − ψ1 (x,t) ,
∂t
x ∈ Ω, t ∈ [0,T],
f ψ1 (x,t)
∂ψ2
− D2 Δψ2 = ψ2 (x,t) − d +
,
∂t
ψ1 (x,t) + ψ2 (x,t)
x ∈ Ω, t ∈ [0,T],
∂ψ1
≥ 0,
∂n
ψ1 (x,t) ≥ 0,
(2.9)
∂ψ2
≥ 0, x ∈ ∂Ω, t ∈ [0,T],
∂n
ψ2 (x,0) ≥ 0, x ∈ Ω̄, t ∈ [−τ,0].
Since a = αe−γτ = 0 and b12 ≡ 0, by Lemma 2.2 we have
(x,t) ∈ Ω̄ × [0,σ].
ui (x,t) ≥ 0,
(2.10)
Note that ψ1 (x,t) is bounded in Ω̄ × [0,σ] for any σ(0 < σ ≤ T). If maxΩ×[0,σ] ψ1 (x,t) ≥
ϕ1 ∞ , due to ψ1 (x,t) satisfying the homogeneous Neumann boundary condition, there
exists (x0 ,t0 ) ∈ Ω × [0,σ] such that
ψ1 (x0 ,t0 ) = max ψ1 (x,t) ≥ ϕ1 ∞ .
Ω×[0,σ]
(2.11)
Therefore, from the first equation of system (2.9) at the point (x0 ,t0 ), we have
aψ1 (x,t − τ) − ψ1 2 (x,t) (x0 ,t0 ) ≥ 0.
(2.12)
That is
ψ1 x0 ,t0 ≤ a.
(2.13)
Hence, we obtain
0 ≤ ψ1 (x,t) ≤ max ϕ1 ∞ ,a ,
(x,t) ∈ Ω × [0,σ].
(2.14)
Xinyu Song et al. 7
Taking the same argument in [σ,2σ], [2σ,3σ],...,[(n − 1)σ, nσ(= T)], we have
0 ≤ ψ1 (x,t) ≤ M1 ,
(x,t) ∈ Ω × [0,T].
(2.15)
Similarly, there exists (x 0 ,t 0 ) ∈ Ω × [0,T] such that
ψ2 (x,t) − d +
f ψ1 (x,t)
ψ1 (x,t) + ψ2 (x,t)
(x 0 ,t 0 )
≥ 0.
(2.16)
Hence, if f > d, then
0 ≤ ψ2 (x,t) ≤
M1 ( f − d)
.
d
(2.17)
i = 1,2.
(2.18)
By Lemma 2.3, we have
ui (x,t) ≤ ψi (x,t),
So we have
0 ≤ u1 (x,t) ≤ max ϕ1 ∞ ,a ,
M1 ( f − d)
0 ≤ u2 (x,t) ≤ max ϕ2 ∞ ,
.
d
(2.19)
3. Local asymptotic stability of the equilibria
In this section, we discuss local asymptotic stability of the nonnegative equilibria by linearization method and analyzing the so-called characteristic equation of the equilibrium.
It is obvious that system (2.1) only has three nonnegative equilibria: the equilibrium
E1 (0,0), the equilibrium E2 (a,0), and the positive equilibrium E3 (c1∗ ,c2∗ ) when f > d and
a/b > 1 − d/ f , where
c1∗ =
(a − b) f + bd
,
f
c2∗ =
( f − d)c1∗
.
d
(3.1)
We will point out that E1 (0,0) cannot be linearized though it is defined for system
(2.1), so the local stability of E1 (0,0) will be studied in another paper.
Let μ1 < μ2 < μ3 < · · · < μn < · · · be the eigenvalues of the operator −Δ on Ω with
the homogeneous Neumann boundary condition, and let E(μi ) be the eigenfunction
space corresponding to μi in C 1 (Ω). It is well known that μ1 = 0 and the correspondbasis of
ing eigenfunction φ1 (x) > 0. Let {φi j | j = 1,2,..., dimE(μi )} be an orthogonal
X
E(μi ), X = {u = (u1 ,u2 ) | u ∈ [C 1 (Ω)]2 } and Xi j = {cφi j | c ∈ R2 }, thus X = ∞
i=1 i , Xi =
dimE(μi )
Xi j .
j =1
8
A ratio-dependent predator-prey system
Let u1 (x,t) = u∗1 (x,t) + c1∗ , u2 (x,t) = u∗2 (x,t) + c2∗ , where c1∗ and c2∗ are both not zero.
We still make u1 (x,t), u2 (x,t) corresponding to u∗1 (x,t), u∗2 (x,t), so the linearized equation of the system (2.1) at (c1∗ ,c2∗ ) is
2
2
b c2∗
b c1∗
∂u1
− D1 Δu1 = au1 (x,t − τ) − 2c1∗ u1 (x,t) − u
(x,t)
−
2 u2 (x,t),
1
2
∂t
c1∗ + c2∗
c1∗ + c2∗
2
2
f c2∗
f c1∗
∂u2
− D2 Δu2 = u
(x,t)
−
du
(x,t)
+
2 u2 (x,t), x ∈ Ω, t > 0,
1
2
2
∂t
c1∗ + c2∗
c1∗ + c2∗
∂u1 ∂u2
=
= 0, x ∈ ∂Ω, t > 0,
∂n
∂n
u1 (x,t) = ϕ1 (x,t) − c1∗ , u2 (x,t) = ϕ2 (x,0) − c2∗ , x ∈ Ω, t ∈ [−τ,0].
(3.2)
From [5], we know that the characteristic equation for the system (3.2) is equivalent
to
2
b c2∗
λ + μk D1 − ae−λτ + 2c1∗ + 2
c1∗ + c2∗
∗ 2
f c2
−
2
c∗ + c∗
1
That is
2
⎛
2
b c1∗
∗
2
c1 + c2∗
∗ 2 = 0.
f c1
λ + μk D2 + d − ∗ ∗ 2 c1 + c2 2 ⎞ ⎛
2 ⎞
∗
∗
⎝λ + μk D1 − ae−λτ + 2c∗ + b c2 ⎠ ⎝λ + μk D2 + d − f c1 ⎠
1
2
2
c1∗ + c2∗
c1∗ + c2∗
⎛
2 ⎞ ⎛
2 ⎞
f c∗
b c∗
+ ⎝ ∗ 1 ∗ 2 ⎠ ⎝ ∗ 2 ∗ 2 ⎠ = 0.
c1 + c2
c1 + c2
(3.3)
(3.4)
3.1. Local asymptotic stability of the equilibrium E2 (a,0). From (3.4), it follows that at
the equilibrium E2 (a,0),
λ + μk D1 − ae−λτ + 2a λ + μk D2 + d − f = 0.
(3.5)
From the first factor of (3.5), we see
λ + μk D1 + 2a = ae−λτ .
(3.6)
λ + μk D1 + 2a = ae−λτ .
(3.7)
Therefore,
Now we will determine that all roots of (3.7) satisfy Reλ < 0. Suppose that there exists λ0
such that Re λ0 ≥ 0. From (3.7), we deduce that
λ0 + μk D1 + 2a ≤ |a| e−τ Reλ0 ≤ |a|.
(3.8)
Xinyu Song et al. 9
This implies that λ0 is in the circle in the complex plane centered at (−(μk D1 + 2a),0)
and of radius a. However, as for given μk and D1 , it follows for ever that μk D1 + 2a > a,
therefore,
Reλ < 0.
(3.9)
λ = −μk D2 − d + f ≤ f − d.
(3.10)
By the second factor of (3.5), we have
If f > d, by taking k = 1(μ1 = 0), from (3.10), we obtain that there at least exists a root λ0
of (3.5) such that Reλ0 > 0. Therefore, E2 (a,0) is unstable if the condition f > d holds.
If f < d, then f − d < 0, by (3.10), we have Reλ < 0. Therefore, if f < d, then E2 (a,0)
is locally asymptotically stable.
3.2. Local asymptotic stability of the equilibrium E3 (c1∗ ,c2∗ ). Let λ = x + iy, using (3.4),
a direct calculation yields
⎛
⎝x + iy + μk D1 − ae−xτ cos(− yτ) + isin(− yτ)
2 ⎞ ⎛
2 ⎞
f c∗
b c∗
+2c1∗ + ∗ 2 ∗ 2 ⎠ ⎝x + iy + μk D2 + d − ∗ 1 ∗ 2 ⎠
c1 + c2
c1 + c2
⎛
2 ⎞ ⎛
(3.11)
2 ⎞
f c∗
b c∗
+ ⎝ ∗ 1 ∗ 2 ⎠ ⎝ ∗ 2 ∗ 2 ⎠ = 0,
c1 + c2
c1 + c2
where c1∗ = ((a − b) f + bd)/ f , c2∗ = (( f − d)c1∗ )/d.
Throughout the section we assume f ≥ 2d and a f ≥ 2b( f − d) and let
2
b c∗
M1 = x + μk D1 − ae cos(− yτ) + 2c1 + ∗ 2 ∗ 2 ,
c1 + c2
M2 = y + ae−xτ sin(yτ),
−xτ
∗
f c1∗
2
(3.12)
N1 = x + μk D2 + d − ∗ ∗ 2 ,
c1 + c2
N2 = y.
Separating real and imaginary parts and applying (3.12) to (3.11), we obtain the equations
⎛
2 ⎞
2 ⎞ ⎛
f c2∗
b c1∗
⎝
⎠
⎝
M1 N1 − M2 N2 + ∗ ∗ 2
∗
⎠ = 0,
∗ 2
(3.13)
M1 N2 + M2 N1 = 0.
(3.14)
c1 + c2
c1 + c2
10
A ratio-dependent predator-prey system
Assume, for contradiction, that there exists a root λ such that Re λ = x ≥ 0. By (3.12), we
have
M1 = x + μk D1 − ae
∗
≥ x + 0 − ae
−xτ
2
b c∗
cos(− yτ) + 2c 1 + ∗ 2 ∗ 2
c1 + c2
−xτ
2
b c∗
cos(− yτ) + 2c1 + ∗ 2 ∗ 2
c1 + c2
∗
2
b c2∗
2
c1∗ + c2∗
≥ x − a + 2c1∗ + (3.15)
2
b c2∗
bc∗
bc2∗
∗
− a + c1 + ∗ 2 ∗ + 2 + c1 − ∗
∗
∗
c1 + c2
c1 + c2∗
c1 + c2
∗
≥
2
2
b c2∗
b c2∗
bc2∗
≥
+
b(
f
−
d)
f
−
=
2 > 0,
2
c1∗ + c2∗
c1∗ + c2∗
c1∗ + c2∗
f c1∗
2
N1 = x + μk D2 + d − ∗ ∗ 2
c1 + c2
2
2
f c1∗
f c1∗
f c1∗
f c1∗
=d− ∗
2 > 0.
∗ + ∗
∗ −
∗
∗ 2 ≥ ∗
c1 + c2 c1 + c2 (c1 + c2 )
c1 + c2∗
(3.16)
Applying (3.15) and (3.16), one can obtain
⎛
∗ 2 ⎞
∗ 2 ⎞ ⎛
⎝ b c2 ⎠ ⎝ f c1 ⎠ ≤ M1 N1 .
∗
∗ 2
∗
∗ 2
c1 + c2
(3.17)
c1 + c2
Using (3.13) and (3.14), we have
⎛⎛
2 ⎞ ⎛
2 ⎞⎞2
∗
∗
⎝⎝ b c1 ⎠ ⎝ f c2 ⎠⎠
2
2
c1∗ + c2∗
c1∗ + c2∗
(3.18)
2 2 2 2
= M2 N 2 + M1 N 1 + M1 N 2 + M2 N 1 .
If N2 = 0, by (3.15) and (3.18), we get
⎛⎛
2 ⎞ ⎛
2 ⎞⎞2
∗
∗
⎝⎝ b c1 ⎠ ⎝ f c2 ⎠⎠
2
2
c1∗ + c2∗
c1∗ + c2∗
= M2 N 2
2
+ M1 N 1
2
+ M1 N 2
(3.19)
2
+ M2 N 1
2
2
> M1 N 1 ,
Xinyu Song et al.
11
it is a contradiction to (3.17). If N2 = 0, from (3.12), we deduce M2 = 0, again using
(3.13), we have
⎛
∗ 2 ⎞
∗ 2 ⎞ ⎛
⎝x + μk D1 − ae−xτ + 2c∗ 1 + b c 2 ⎠ ⎝x + μk D2 + d − f c1 ⎠
2
2
⎛
⎞⎛
2
c1∗ + c2∗
2
c1∗ + c2∗
⎞
(3.20)
f c∗
b c∗
+ ⎝ ∗ 1 ∗ 2 ⎠ ⎝ ∗ 2 ∗ 2 ⎠ = 0,
c1 + c2
c1 + c2
that is,
⎛
⎞⎛
⎞
∗ ∗
∗ ∗
⎝x + μk D2 + f c1 c2 ⎠ ⎝x + μk D1 + a − ae−xτ + c∗ − bc1 c2 ⎠
1
2
2
c1∗ + c2∗
c1∗ + c2∗
⎛
2 ⎞ ⎛
2 ⎞
f c∗
b c∗
+ ⎝ ∗ 1 ∗ 2 ⎠ ⎝ ∗ 2 ∗ 2 ⎠ = 0.
c1 + c2
c1 + c2
(3.21)
It is obvious that x = −μk D2 − f c1∗ c2∗ /(c1∗ + c2∗ )2 does not satisfy (3.21), so we have
⎛
⎞
∗ ∗
⎝x + μk D2 + f c1 c2 ⎠
2
c1∗ + c2∗
⎛
bc1∗ c2∗
2
c1∗ + c2∗
× ⎝x + μk D1 + a − ae−xτ + c1∗ − +
2 b c1∗ / c1∗ + c2∗
2 (3.22)
2 f c2∗ / c1∗ + c2∗
x + μk D2 + f c1∗ c2∗ / c1∗ + c∗ 2
2
2 ⎞
⎟
⎠ = 0.
So all roots of (3.22) are given by (3.23), that is,
2 2 2 b c1∗ / c1∗ + c2∗
f c2∗ / c1∗ + c2∗
bc∗ c∗
x = −μk D1 − a − c1∗ + ae−xτ + ∗ 1 2∗ 2 −
2
c1 + c2
x + μk D2 + f c1∗ c2∗ / c1∗ + c2∗
2 2 2 b c1∗ / c1∗ + c2∗
f c2∗ / c1∗ + c2∗
bc∗ c∗
≤ −c1∗ + 1 2 2 −
2
c1∗ + c2∗
x + μk D2 + f c1∗ c2∗ / c1∗ + c2∗
x + μk D2
bc∗ c∗
≤ −c1 + 1 2 2
2
∗
∗
c1 + c2
x + μk D2 + f c1∗ c2∗ / c1∗ + c2∗
2 2 ∗
bc∗ c∗
b
b
< −c1∗ + ∗ 1 2∗ 2 < c1∗ − 1 + ∗ ∗ ≤ c1∗ − 1 + ∗ ≤ 0,
c
+
c
2c
c1 + c2
1
2
1
(3.23)
it is a contradiction to Re λ = x ≥ 0. So we have that Reλ < 0 if f ≥ 2d and a f ≥ 2b( f − d),
that is, the positive equilibrium E3 (c1∗ ,c2∗ ) is locally asymptotically stable.
12
A ratio-dependent predator-prey system
From the above discussion, we can conclude the following.
Theorem 3.1. If f ≥ 2d and a f ≥ 2b( f − d), then the positive equilibrium E3 (c1∗ ,c2∗ ) is
locally asymptotically stable.
Theorem 3.2. If f > d, then the equilibrium E2 (a,0) is unstable.
Theorem 3.3. If f < d, then the equilibrium E2 (a,0) is locally asymptotically stable.
4. Global asymptotic stability of the equilibria
Note that F1 (u1 (x,t),u2 (x,t),u1 (x,t − τ)) and F2 (u1 (x,t),u2 (x,t)), with respect to u1 , u2 ,
are continuous and mixed quasimonotone in Σ × Σ∗ , where Σ, Σ∗ are fixed and bounded
subsets of R2 . Thus there exist Ki ≥ 0 (i = 1,2) such that
Fi u1 ,u2 ,u1 (x,t − τ) − Fi u 1 ,u 2 ,u 1 (x,t − τ) (4.1)
≤ Ki u1 (x,t) − u 1 (x,t) + u2 (x,t) − u 2 (x,t) ,
when i = 2, τ = 0, where (u1 ,u 1 ),(u2 ,u 2 ) ∈ Σ × Σ∗ .
In order to investigate the dynamics of the system (2.1) we define two sequences of
(m) } = {c (m) ,c (m) }∞
constant vectors {c̄(m) } = {c¯1 (m) , c¯2 (m) }∞
m=1 , {c
m=1 satisfying the fol1
2
lowing relation:
c¯1
(m)
c1
(m)
c¯2
(m)
c2
(m)
= c¯1
(m−1)
bc (m−1)
1
+ c¯1 (m−1) a − c¯1 (m−1) − (m) 2 (m−1) ,
K1
c¯1 + c2
= c1
(m−1)
1
bc¯ (m−1)
+ c1 (m−1) a − c1 (m−1) − (m−1)2
,
K1
c1
+ c¯2 (m−1)
= c¯2
(m−1)
f c¯1 (m)
1
+ c¯2 (m−1) − d + (m)
,
K2
c¯1 + c¯2 (m)
= c2
(m−1)
f c (m−1)
1
+ c2 (m−1) − d + (m−1)2
,
K2
c2
+ c2 (m−1)
c̄(0) = c,
c(0) = c,
(4.2)
m = 1,2,...,
where (c, c) is a pair of coupled upper and lower solutions of system (2.1). It is easy to
prove the following lemma.
Lemma 4.1. The sequences {c̄(m) }, {c(m) } given by (4.2) with c̄(0) = c and c(0) = c possess
the monotone property
c ≤ c(m) ≤ c(m+1) ≤ c̄(m+1) ≤ c̄(m) ≤ c,
m = 1,2,....
(4.3)
Xinyu Song et al.
13
Proof. Since (c, c) is a pair of coupled upper and lower solutions of the system (2.1), so it
follows from (2.2) that
(0)
c¯1
− c¯1
(1)
= c¯1
(0)
− c¯1
(0)
bc (0)
1
+ c¯1 (0) a − c¯1 (0) − (0) 2 (0)
K1
c¯1 + c2
bc (0)
1
= − c¯1 (0) a − c¯1 (0) − (0) 2 (0)
K1
c¯1 + c2
c1
(1)
− c1
(0)
= c1
≥ 0,
1
bc¯ (0)
+ c1 (0) a − c1 (0) − (0) 2 (0)
K1
c1 + c¯2
1 (0)
bc¯ (0)
=
c1
a − c1 (0) − (0) 2 (0)
K1
c1 + c¯2
(4.4)
(0)
− c1 (0)
(4.5)
≥ 0.
This gives c¯1 (0) ≥ c¯1 (1) and c1 (1) ≥ c1 (0) . Similarly by (2.2) and the quasimonotone property, we have
K1 c¯1 (1) − c1 (1) = K1 c¯1 (0) − c1 (0) + c¯1 (0) a − c¯1 (0) −
− c1
(0)
a − c1
bc¯2 (0)
− (0)
c1 + c¯2 (0)
(0)
bc2 (0)
(0)
c¯1 + c2 (0)
(4.6)
≥ 0.
This yields c¯1 (1) ≥ c1 (1) . The above conclusions show that
c(0) ≤ c(1) ≤ c̄(1) ≤ c̄(0) .
(4.7)
Assume, by induction, that c(m−1) ≤ c(m) ≤ c̄(m) ≤ c̄(m−1) for some m > 1, Then by (4.6)
and (2.2), we have
K1 c¯1 (m) − c1 (m+1)
= K1 c¯1
(m−1)
− c1
(m)
− c1
(m)
a − c1
(m)
+ c¯1
(m−1)
a − c¯1
bc¯2 (m)
− (m)
c1 + c¯2 (m)
(m−1)
bc (m−1)
− (m−1)2
+ c2 (m−1)
c¯1
(4.8)
≥ 0.
This yields c¯1 (m) ≥ c1 (m+1) . A similar argument gives c(m) ≤ c(m+1) ≤ c̄(m+1) ≤ c̄(m) . A similar argument gives c¯2 (m) and c2 (m) . Therefore, the monotone property (4.3) follows by the
principle of induction.
By monotone bounds principle, we get
lim c(m) = c,
m→∞
lim c̄(m) = c̄,
m→∞
(4.9)
14
A ratio-dependent predator-prey system
with
c ≤ c(m) ≤ c(m+1) ≤ c ≤ c̄ ≤ c̄(m+1) ≤ c̄(m) ≤ c.
(4.10)
In (4.2), letting m → ∞, we have
c¯1 a − c¯1 −
c1 a − c1 −
f c¯1
= 0,
c¯1 + c¯2
c2 − d +
c¯2 − d +
bc2
= 0,
¯
c1 + c2
bc¯2
= 0,
c1 + c¯2
f c1
= 0.
c1 + c2
(4.11)
Considering [16, Theorem 2.2] as a corollary, we obtain the following.
Theorem 4.2. Let c, c̄ be the limits in (4.2), then for any initial function ϕ = (ϕ1 ,ϕ2 ) in
c, c̄ the solution of system (2.1) satisfies the relation
c ≤ u(x,t) ≤ c̄ as t → ∞.
(4.12)
Moreover, if, in addition, c = c̄(≡ c∗ ), then c∗ is the unique global solution of the system
(2.1) in c, c̄, and
lim u(x,t) = c∗ ,
x ∈ Ω.
t →∞
(4.13)
Theorem 4.3. Let (u1 (x,t),u2 (x,t)) be the solution of the system (2.1), if ϕ1 (x,0) = 0 and
ϕ2 (x,0) ≡ 0, then u1 (x,t) > 0, u2 (x,t) ≡ 0, and
u1 (x,t),u2 (x,t) −→ (a,0),
t −→ +∞.
(4.14)
Proof. By the standard maximum principle for parabolic boundary-value problems with
homogeneous Neumann boundary condition, we see that
u1 (x,t),u2 (x,t) ≥ (0,0).
(4.15)
It is obvious that if ϕ1 (x,0) = 0, ϕ2 (x,0) ≡ 0, then
u1 (x,t) > 0,
u2 (x,t) ≡ 0.
(4.16)
Now, the system (2.1) becomes the scalar boundary-value problem
∂u1
− D1 Δu1 = au1 (x,t − τ) − u1 2 (x,t),
∂t
∂u1
= 0,
∂n
x ∈ ∂Ω, t > 0;
u1 (x,t) = ϕ1 (x,t) ≥ 0,
x ∈ Ω, t > 0,
(4.17)
x ∈ Ω, t ∈ [−τ,0].
By continuity and Lemma 2.3, it follows that there exist δ > 0 and t ∗ > 0 satisfying
u1 (x,t) ≥ δ,
(x,t) ∈ [t ∗ ,t ∗ + τ].
(4.18)
Xinyu Song et al.
15
Let c = M, c = ε ≥ 0, where
M = max
max
∗ ∗
Ω×[t ,t +τ]
ε = min{a,δ }.
u1 (x,t),a ,
(4.19)
Then
ε ≤ a =⇒ ε2 ≤ aε =⇒ aε − ε2 ≥ 0,
a ≤ M =⇒ aM ≤ M 2 =⇒ aM − M 2 ≤ 0.
(4.20)
Therefore, (c, c) is a pair of strong upper and lower solutions of the system (4.17). Applying Theorem 4.2 and (4.3), we obtain that there exist c, c̄ such that
ε ≤ c ≤ c ≤ c̄ ≤ c ≤ M,
c(a − c) = 0,
c̄(a − c̄) = 0,
(4.21)
while the equation c(a − c) = 0 has a unique positive solution c∗ = a.
Therefore
c̄ = c = c∗ = a,
lim u1 (x,t) = c∗ = a.
t →∞
(4.22)
Theorem 4.4. If f ≥ 2d and a f ≥ 2b( f − d), then for any nonnegative initial function
ϕ = (ϕ1 ,ϕ2 ) the system (2.1) has a unique global solution with
u1 (x,t),u2 (x,t) −→ c1 ∗ ,c2 ∗
as t −→ +∞.
(4.23)
Proof. Let ω1 (x,t) be the solution of the scalar boundary-value problem
∂ω1
− D1 Δω1 = aω1 (x,t − τ) − ω1 2 (x,t), x ∈ Ω, t > 0,
∂t
∂ω1
= 0, x ∈ ∂Ω, t > 0,
∂n
ω1 (x,t) = ϕ1 (x,t) ≥ 0, x ∈ Ω, t ∈ [−τ,0].
(4.24)
Applying Lemma 2.3, we have
u1 (x,t) ≤ ω1 (x,t),
(x,t) ∈ Ω × [0,+∞].
(4.25)
By Theorem 4.3, it follows that
ω1 (x,t) −→ a,
t −→ ∞.
(4.26)
Hence, for any ε > 0, there always exists t ∗∗ > 0 as t > t ∗∗ such that
u1 (x,t) ≤ ω1 (x,t) ≤ a + ε.
(4.27)
16
A ratio-dependent predator-prey system
Let ω2 (x,t) be the solution of the scalar boundary-value problem
f (a + ε)ω2 (x,t)
∂ω2
− D2 Δω2 = −dω2 (x,t) +
∂t
(a + ε) + ω2 (x,t)
∂ω2
= 0,
∂n
(x,t) ∈ ∂Ω × t
(x,t) ∈ Ω × t ∗∗ ,+∞ ,
≤ −dω2 (x,t) + f (a + ε),
ω2 (x,t) = ϕ1 x,t ∗∗ ≥ 0,
∗∗
(4.28)
,+∞ ,
x ∈ Ω.
By the quasimonotone property of F2 (u1 ,u2 ), we obtain
u2 (x,t) ≤ ω2 (x,t),
(x,t) ∈ Ω × [t ∗∗ ,+∞].
(4.29)
Now we consider the scalar boundary-value problem
∂ω2
− D2 Δω2 = −dω2 (x,t) + f (a + ε), (x,t) ∈ Ω × [t ∗∗ ,+∞],
∂t
∂ω2
= 0, (x,t) ∈ ∂Ω × [t ∗∗ ,+∞],
∂n
ω2 (x,t) = ϕ1 (x,t ∗∗ ) ≥ 0, x ∈ Ω.
(4.30)
By [22], we have ω2 (x,t) → f (a + ε)/d as t sufficiently large. Therefore, for the above given
ε, there exists t0 > 0 satisfying
u2 (x,t) ≤
f (a + ε)
,
d
(x,t) ∈ Ω × t0 ,+∞ .
(4.31)
Let
c1 = a + ε,
c2 =
f (a + ε)
,
d
c
1 = δ,
(4.32)
c
2 = δ,
where ε and δ are sufficiently small positive constants.
By f ≥ 2d and a f ≥ 2b( f − d), applying Lemmas 2.2 and 2.3, for t sufficiently large,
we obtain that
c
1 ≤ u1 (x,t) ≤ c1 ,
c2 =
c
2 ≤ u2 (x,t) ≤ c2 ,
f (a + ε)
f (a + ε)c2
f (a + ε)c2
=⇒ d c2 = f (a + ε) =⇒ d c2 ≥
=⇒ −d c2 +
≤ 0,
d
c1 + c2
c1 + c2
2
2
2
c1 = a + ε ≥ a =⇒ c1 ≥ ac1 =⇒ 0 ≥ ac1 − c1 =⇒ ac1 − c1 −
bδ c1
≤ 0.
δ + c1
(4.33)
Let ε and δ be sufficiently small positive constants. Applying f ≥ 2d and a f ≥ 2b( f − d),
it is not difficult to prove that c
1 = δ1 and c
2 = δ2 satisfy (2.2).
Xinyu Song et al.
17
Hence, (c1 , c2 ), (c
1 , c
2 ) are a pair of coupled upper and lower solutions of the system
(2.1).
By Theorem 4.2, we obtain that c and c̄ satisfy
0 < c ≤ c ≤ c̄ ≤ c,
c¯1 a − c¯1 −
c1 a − c1 −
bc¯2
= 0,
c1 + c¯2
(4.34)
c¯2 − d +
c2
bc2
= 0,
c¯1 + c2
f c¯1
= 0,
c¯1 + c¯2
f c1
−d+
= 0.
c1 + c2
Since f ≥ 2d and a f ≥ 2b( f − d) and the following equations have unique positive solutions:
c1 a − c1 −
bc2
= 0,
c1 + c2
c2 − d +
f c1
= 0,
c1 + c2
(4.35)
therefore
c¯1 = c1 = c1 ∗ ,
c¯2 = c2 = c2 ∗ ,
(4.36)
where
c∗ 1 =
(a − b) f + bd
,
f
c∗ 2 =
( f − d)c∗ 1
.
d
(4.37)
Therefore
lim u1 (x,t) = c1∗ ,
lim u2 (x,t) = c2∗ .
t →+∞
t →+∞
(4.38)
Theorem 4.5. If f < d, then for any nonnegative initial function ϕi (x,t) (i = 1,2), the
system (2.1) has a unique global nonnegative solution (u1 (x,t),u2 (x,t)) satisfying
u1 (x,t),u2 (x,t) −→ (a,0),
t −→ +∞.
(4.39)
Proof. Let c1 ∗ and c2 ∗ be the solutions of the following system:
c1
∗
a − c1
c2 ∗
∗
bc2 ∗
− ∗
c1 + c2 ∗
f c1 ∗
−d+ ∗
c1 + c2 ∗
= 0,
(4.40)
= 0.
18
A ratio-dependent predator-prey system
If f < d, then (4.40) has only one nonzero solution and one nonnegative solution:
c1 ∗ = a,
c2 ∗ = 0.
(4.41)
Let c2 be sufficiently large and let 0 < δ(≤ a) be sufficiently small, and c1 , c
1 , c
2 satisfy
c
1 = δ,
c1 ≥ a,
c
2 = 0.
(4.42)
Applying Lemmas 2.2 and 2.3, if the condition f < d holds, it is easy to prove
c
1 ≤ u1 (x,t) ≤ c1 ,
c
2 ≤ u2 (x,t) ≤ c2 ,
(4.43)
and (c1 , c2 ), (c
1 , c
2 ) satisfying (2.2). Hence, (c1 , c2 ), (c
1 , c
2 ) are a pair of coupled upper and
lower solutions of the system (2.1). By Theorem 4.2, we obtain c and c̄ satisfying
0 < c ≤ c ≤ c̄ ≤ c,
bc2
c¯1 a − c¯1 −
= 0,
c¯1 + c2
c1 a − c1 −
c¯2 − d +
c2 − d +
bc¯2
= 0,
c1 + c¯2
(4.44)
f c¯1
= 0,
c¯1 + c¯2
f c2
= 0.
c1 + c2
We see from c2 (0) = 0 that c2 (m) = 0 for every m = 1,2,.... This implies c2 = 0.
Using 0 < δ ≤ c1 ≤ c¯1 , we have c1 = a. Since f < d, and
c¯1 a − c¯1 −
c1
bc2
c¯1 + c2
bc¯2
a − c1 −
c1 + c¯2
c¯2 − d +
f c¯2
c¯1 + c¯2
= 0,
= 0,
(4.45)
= 0,
so we have
c¯1 = c1 = a,
c¯2 = c2 = 0.
(4.46)
By Theorem 4.2, for any nonnegative initial function ϕi (x,t), i = 1,2, the system (2.1) has
a unique global nonnegative solution (u1 (x,t),u2 (x,t)) in ϕi (x,t) ∈ c, c satisfying
(u1 (x,t),u2 (x,t)) −→ (a,0),
t −→ +∞.
(4.47)
Xinyu Song et al.
19
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No.
10471117), the Henan Innovation Project for the University Prominent Research Talents
(No. 2005KYCX017), and the Scientific Research Foundation for the Returned Overseas
Chinese Scholars, State Education Ministry. We would like to thank the referees and the
editor for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improved the presentation of this work.
References
[1] W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure,
Mathematical Biosciences 101 (1990), no. 2, 139–153.
[2] W. G. Aiello, H. I. Freedman, and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM Journal on Applied Mathematics 52 (1992),
no. 3, 855–869.
[3] J. F. M. Al-Omari and S. A. Gourley, Stability and traveling fronts in Lotka-Volterra competition
models with stage structure, SIAM Journal on Applied Mathematics 63 (2003), no. 6, 2063–2086.
[4] H. Amann, Dynamic theory of quasilinear parabolic equations II: reaction-diffusion systems, Differential and Integral Equations 3 (1990), no. 1, 13–75.
[5] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion,
Journal of Mathematical Analysis and Applications 254 (2001), no. 2, 433–463.
[6] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and
Textbooks in Pure and Applied Mathematics, vol. 57, Marcel Dekker, New York, 1980.
[7] H. I. Freedman and J. Wu, Persistence and global asymptotic stability of single species dispersal
models with stage structure, Quarterly of Applied Mathematics 49 (1991), no. 2, 351–371.
[8] S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology 49 (2004), no. 2, 188–200.
[9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics
in Science and Engineering, vol. 191, Academic Press, Massachusetts, 1993.
[10] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,
Journal of Mathematical Biology 36 (1998), no. 4, 389–406.
[11] J. D. Murray, Mathematical Biology, Biomathematics, vol. 19, Springer, Berlin, 1989.
[12] P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model,
Journal of Differential Equations 200 (2004), no. 2, 245–273.
[13] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
, Dynamics of nonlinear parabolic systems with time delays, Journal of Mathematical Anal[14]
ysis and Applications 198 (1996), no. 3, 751–779.
, Systems of parabolic equations with continuous and discrete delays, Journal of Mathe[15]
matical Analysis and Applications 205 (1997), no. 1, 157–185.
, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal[16]
ysis. Series A: Theory and Methods 48 (2002), no. 3, 349–362.
[17] S. G. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, The Canadian Applied
Mathematics Quarterly 2 (1994), no. 4, 485–550.
[18] J. W.-H. So, J. Wu, and X. Zou, A reaction-diffusion model for a single species with age structure.
I. Traveling fronts on unbounded domains, Proceedings of the Royal Society of London. Series A
457 (2001), no. 2012, 1841–1853.
[19] X. Song, L. Cai, and A. U. Neumann, Ratio-dependent predator-prey system with stage structure
for prey, Discrete and Continuous Dynamical Systems. Series B 4 (2004), no. 3, 747–758.
20
A ratio-dependent predator-prey system
[20] X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Mathematical Application Sinica 18 (2002), no. 3, 307–314.
[21] Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, New Jersey,
1996.
[22] M. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, 1993.
Xinyu Song: Department of Mathematics, Xinyang Normal University, Xinyang 464000,
Henan, China
E-mail address: xysong88@163.com
Zhihao Ge: Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China
E-mail address: zhihaoge@gmail.com
Jingang Wu: Department of Mathematics, Xinyang Normal University, Xinyang 464000,
Henan, China
E-mail address: kyczrk@mail2.xytc.edu.cn